Dual (category theory)

(Redirected from Opposite category)

In category theory, an abstract branch of mathematics, the dual of a category C is the category formed by reversing all the morphisms of C. That is, we take Cop to be the category with objects that are those of C, but with the morphisms from X to Y in Cop being the morphisms from Y to X in C. Hence, the dual of a dual of a category is itself. The dual category is also called the opposite category.

Examples come from reversing the direction of inequalities in a partial order. So if X is a set and ≤ a partial order relation, we can define a new ≤new by the definition

xnew y if and only if yx.

For example, there are opposite pairs child/parent, or descendant/ancestor.

This is a special case, since partial orders correspond to a certain kind of category in which Mor(A,B) can have at most one element. In applications to logic, this then looks like a very general description of negation (that is, proofs run in the opposite direction). For example, if we take the opposite of a lattice, we will find that meets and joins have their roles interchanged. This is an abstract form of De Morgan's laws.

Generalising that observation, limits and colimits are interchanged when one passes to the opposite category. This is immediately useful, when one can identify the opposite category in concrete terms. For example the category of affine schemes is equivalent to the opposite of the category of commutative rings. The Pontryagin duality restricts to an equivalence between the category of compact Hausdorff abelian topological groups and the opposite of the category of (discrete) abelian groups. The category of Stone spaces and continuous functions is equivalent to the opposite of the category of Boolean algebras and homomorphisms.

A duality between categories C and D is defined as an equivalence between C and the opposite of D. The above are all examples of dualities.

One other way in which the concept is used is to remove the distinction between covariant and contravariant functors: a contravariant functor to D is equally a functor to the opposite of D.

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